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Lie derivations on Banach algebras. (English) Zbl 0957.46032
A Lie derivation $\Delta$ on a unital complex Banach algebra $A$ is considered. The main result of the article is the following Theorem. Let $D$ be a Lie derivation on a unital complex Banach algebra $A$. Then for every primitive ideal $P$ of $A$, except for a finite set of them which have finite codimension greater than one, there exist a derivation $d$ from $A/P$ to itself and a linear functional $\tau$ on $A$ such that $$Q_p\Delta(a)= d(a+ P)+ \tau(a)$$ for all $a\in A$ (where $Q_p$ denotes the quotient map from $A$ onto $A(p)$. Moreover, the preceding decomposition holds for all primitive ideals in the case where $\Delta$ is continuous.

46H05General theory of topological algebras
Full Text: DOI
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